On the distribution modulo 1 of the sequence $w(n) e n!$

Articles in Press

Document Type : Research Paper

Authors
Mohammad Ghorbani 1
Mehdi Hassani 1
Hossein Moshtagh‎ 2
1 Department of Mathematics, Faculty of Science, University of Zanjan, University Blvd., 45371-38791, Zanjan, Iran
2 Department of Computer science, University of Garmsar, 35881-15589, Garmsar, Semnan, Iran
10.22075/ijnaa.2025.36265.5354
Abstract
In this paper, we study the distribution modulo 1 of sequences of the form $a_n=w(n)\lbrace\mathrm{e} n!\rbrace$. We show that if the weight function $w$ is a nonlinear polynomial with a leading irrational coefficient, the $a_n$ is uniformly distributed modulo 1. Also, we show that if $w(n)=\alpha n+\beta$ (linear polynomial),  then $w(n)\lbrace\mathrm{e} n!\rbrace\to\alpha$ as $n\to\infty$, and if $w(n)=O(n^{1-\epsilon})$ (sublinear growth) for some $\epsilon>0$, then $w(n)\lbrace\mathrm{e} n!\rbrace\to 0$ as $n\to\infty$.
Keywords
fractional part
distribution modulo 1
asymptotic expansion
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International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 24 July 2026

  • Receive Date 15 December 2024
  • Revise Date 10 August 2025
  • Accept Date 11 August 2025